Additive disequilibrium (D) is a statistic that estimates the difference between observed genotypic frequencies and the genotypic frequencies that would be expected under Hardy–Weinberg equilibrium. At a biallelic locus with alleles 1 and 2, the additive disequilibrium exists according to the equations
\begin{align} f11&=
2 | |
p | |
1 |
+D\\[5pt] f12&=2p1(1-p1)-2D\\[5pt] f22&=
2 | |
(1-p | |
1) |
+D \end{align}
where fij is the frequency of genotype ij in the population, p is the allele frequency in the population, and D is the additive disequilibrium coefficient.
Having a value of D > 0 indicates an excess of homozygotes/deficiency of heterozygotes in the population, whereas D < 0 indicates an excess of heterozygotes/deficiency of homozygotes. When D = 0, the genotypes are considered to be in Hardy Weinberg Equilibrium. In practice, the estimated additive disequilibrium from a sample,
\widehat{D}
Because the genotype and allele frequencies must be positive numbers in the interval (0,1), there exists a constraint on the range of possible values for D, which is as follows:
maxu\in
2 | |
-p | |
u |
\leD\lep1(1-p1)
To estimate D from a sample, use the formula:
\widehat{D}=\widehat{f}11-
2 | |
\widehat{p} | |
1 |
=
n11 | |
n |
-\left(
2n11+n12 | |
2n |
\right)2
where n11 (n12) is the number of individuals in the sample with that particular genotype and n is the total number of individuals in the sample. Note that
\widehat{f}11
\widehat{p}1
The approximate sampling variance of
\widehat{D}
\operatorname{var}(\widehat{D})
\operatorname{var}\widehatD=
| |||||||||||||||
n |
From this an estimated 95% confidence interval can be calculated, which is
\widehat{D}\pm1.96\sqrt{\operatorname{var}(\widehat{D})}
Note:
\sqrt{\operatorname{var}(\widehat{D})}
If the confidence interval for
\widehat{D}
\widehat{D}
z=
\widehat{D | |
\sqrt |
n}{\widehat{p}1(1-\widehat{p}1)}
When z is large,
\widehat{D}
To determine if z is significantly larger or smaller than expected under Hardy Weinberg Equilibrium, find "the probability of observing" a value as or more extreme as the observed z "under the null hypothesis". The tail probability is normally used,
P
P
If z is negative, find the negative tail probability,
P
The probability values calculated from these equations can be analyzed by comparison to a pre-specified value of α. When the observed probability p ≤ α, we can "reject the null hypothesis of Hardy Weinberg Equilibrium". If p > α, we fail to reject the null hypothesis. Commonly used values of α are 0.05, 0.01, and 0.001.[3]
At a significance of α = 0.05, we can reject the hypothesis of Hardy Weinberg Equilibrium if the absolute value of z is "greater than or equal to the critical value 1.96" for the two-sided test.[4]